integral from x to 2x of y/(x^2+y^2)

Lösung

\int_{x}^{2 x} \frac{y}{x ^{2} + y ^{2}} d y

\frac{1}{2} (ln (5) - ln (2))

Dezimale

0.45814 \dots

Schritte zur Lösung

\int_{x}^{2 x} \frac{y}{x ^{2} + y ^{2}} d y

Wende U-Substitution an

= \int_{2 x^{2}}^{5 x^{2}} \frac{1}{2 u} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{2 x^{2}}^{5 x^{2}} \frac{1}{u} d u

Nutze das gemeinsame Integral :

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} [ln ∣ u ∣]_{2 x^{2}}^{5 x^{2}}

Berechne die Grenzen:

ln (5) - ln (2)

= \frac{1}{2} (ln (5) - ln (2))

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